Serre's 1971 cohomology program for discrete and S-arithmetic groups: finiteness properties, Euler-Poincare characteristics, analytic versus continuous cohomology of p-adic groups, and integrality of zeta values

In his 1971 survey 'Cohomologie des groupes discrets' and the companion Bourbaki expose 399, Jean-Pierre Serre posed five linked questions extending the classical Euler-Poincare package (finite free resolutions, an Euler-Poincare characteristic given by a volume integral, and integrality of that characteristic expressed through special values of Dedekind zeta functions) from compact settings to discrete and S-arithmetic groups in general: (1) does type FP imply type FL for a discrete group; (2) is the Euler-Poincare characteristic of a torsion-free nonuniform lattice of finite covolume given by the Euler-Poincare measure of the quotient, with the group of type FL; (3) does Lazard's comparison between continuous and analytic cohomology extend to noncompact p-adic analytic groups such as SL_2(Q_p); (4) what cohomology and Euler-Poincare theory exists for function-field S-arithmetic groups, which are not virtually torsion-free; (5) are the values w_{2i} zeta_k(1-2i) individually integers, with Kummer-type congruences? (1) was refuted by Bestvina and Brady in 1997; the arithmetic cases of (2) and the finiteness part of (4) are settled (Harder 1971; Borel-Serre 1973, 1976; Behr 1998; Bux-Kohl-Witzel 2013); the odd-primary part of (5) is settled for totally real fields (Wiles 1990, plus the Rost-Voevodsky norm residue theorem); and the comparison program of (3) is active (Casselman-Wigner 1974 and later work). The surviving open core - the general lattice case of (2), the precise noncompact comparison in (3), an Euler-Poincare theory for p-torsion function-field S-arithmetic groups in (4), and the 2-primary and general-field parts of (5) - remains a coherent open program.

Background

Let $\Gamma$ be a discrete group and $\mathbb{Z}\Gamma$ its integral group ring, on which $\Gamma$ acts by left multiplication so that the trivial module $\mathbb{Z}$ carries the constant action. Serre's survey studies the cohomology $H^*(\Gamma,M)$, i.e. the derived functors $\mathrm{Ext}^*_{\mathbb{Z}\Gamma}(\mathbb{Z},M)$, through finiteness properties of $\Gamma$: the group is of type FP if $\mathbb{Z}$ admits a finite resolution by finitely generated projective $\mathbb{Z}\Gamma$-modules, of type FL if the resolution can be chosen with finitely generated free modules, and of type F if it admits a compact classifying complex (a CW-complex with fundamental group $\Gamma$ and contractible universal cover); one has F $\Rightarrow$ FL $\Rightarrow$ FP, and type FL already forces finite presentability, since the first terms of a finite free resolution encode a finite presentation. For a group of type FL the Euler-Poincare characteristic is $\chi(\Gamma)=\sum_i(-1)^i\,\mathrm{rank}_{\mathbb{Z}\Gamma}F_i$ computed from any finite free resolution $F_\bullet\to\mathbb{Z}$, and it is independent of the chosen resolution.

The driving examples are arithmetic groups - subgroups of finite index in the integer points $G(\mathbb{Z})$ of a semisimple algebraic $\mathbb{Q}$-group $G$ - and, more generally, S-arithmetic groups such as $\mathrm{SL}_n(\mathbb{Z}[1/p])$, which are obtained by inverting a finite set $S$ of primes. Such a group acts on the associated symmetric space $G(\mathbb{R})/K$ (and, for each $p$-adic place in $S$, on its Bruhat-Tits building, the contractible polyhedral complex on which $G(\mathbb{Q}_p)$ acts with compact open point stabilizers, apartments being affine Coxeter complexes). The quotient has finite volume but is noncompact in the nonuniform case. Harder's Gauss-Bonnet theorem of 1971 computes $\chi(\Gamma)$ for arithmetically defined groups without compactness as the integral of an Euler-Poincare differential form over the quotient, and expresses it as a product of special values $\zeta_k(1-2i)$ of the Dedekind zeta function of the underlying number field $k$. Two of Serre's questions ask how far this extends: homologically (does FP imply FL?) and geometrically (does the volume-integral formula for $\chi(\Gamma)$, and type FL itself, hold for torsion-free nonuniform lattices - discrete subgroups of finite covolume with noncompact quotient - in general locally compact groups, not only arithmetically defined ones?).

Two further questions concern positive characteristic and p-adic analysis. Over a global function field (a finite extension of $\mathbb{F}_q(t)$), S-arithmetic groups are discrete subgroups of products of groups over local fields of positive characteristic, and they are never virtually torsion-free: every subgroup of finite index contains unipotent elements of order $p$, so the standard device of passing to a torsion-free finite-index subgroup fails, and no Euler-Poincare characteristic is available by the classical route. Separately, for compact p-adic analytic groups Lazard had proved that continuous cohomology (the cohomology computed from cochains that are continuous maps on the topological group) with rational coefficients agrees with the cohomology of the associated p-adic Lie algebra; Serre asked whether a comparison of this kind survives for noncompact p-adic analytic groups such as $\mathrm{SL}_2(\mathbb{Q}_p)$, proposing to compute it through the action on the Bruhat-Tits building, whose compact open stabilizers are within Lazard's reach.

Finally, the integrality of the constants in these formulas: $\zeta_k(s)$ is the Dedekind zeta function of a number field $k$, and $w_i(k)$ denotes the order of the finite group of Galois-invariant elements in the $i$-fold tensor power of the roots of unity of a separable closure of $k$ (for $k=\mathbb{Q}$: $w_2=24$, $w_4=240$, $w_6=504$); the Lichtenbaum normalization pairs $w_{2i}(k)$ with the special value $\zeta_k(1-2i)$. By Euler's formula and the functional equation, $\zeta_{\mathbb{Q}}(1-2k)=-B_{2k}/2k$ with $B_{2k}$ the Bernoulli number of index $2k$, and $\zeta_k$ vanishes at every odd negative integer as soon as $k$ has a complex place, so the question is substantive for totally real $k$ and individual integrality of $w_{2i}\zeta_k(1-2i)$ is a delicate statement; the case $i=1$ is the Birch-Tate conjecture, which relates $\zeta_F(-1)$ for totally real $F$ to the order of $K_2(O_F)$, the second algebraic K-group of the ring of integers $O_F$.

Problem Statement

The source poses five linked questions, whose common objective is to extend the Euler-Poincare package - finite free resolutions, a characteristic computed as a volume integral, and integrality of that characteristic via zeta values - from the classical compact or uniform arithmetic settings to discrete and S-arithmetic groups in general.

(1) Finiteness hierarchy. For a discrete group, does type FP imply type FL? (This component is settled negatively: Bestvina-Brady 1997 groups are of type FP but not finitely presented, hence not of type FL.)

(2) Nonuniform lattices. For a torsion-free lattice $\Gamma$ of finite covolume with noncompact quotient in a general locally compact group $G$ that admits an Euler-Poincare measure $\mu$ (the invariant measure built from the Chern-Weil/Euler form that computes $\chi$ of uniform lattices), is $\chi(\Gamma)=\mu(G/\Gamma)$, and is $\Gamma$ of type FL? The arithmetically defined case is Harder's Gauss-Bonnet theorem; the question is the general one, including non-arithmetic lattices in real rank-one Lie groups.

(3) p-adic comparison. For a noncompact p-adic analytic group $G$, for instance $\mathrm{SL}_2(\mathbb{Q}_p)$, is the continuous cohomology of $G$ with suitable topological coefficients computable by analytic (Lie-algebra) cohomology in the way Lazard established for compact groups - and what exactly is the resulting comparison statement?

(4) Function-field S-arithmetic groups. Develop a cohomology theory with an Euler-Poincare characteristic (and integrality statements for it) for S-arithmetic groups over global function fields, where every finite-index subgroup contains elements of order $p$ and no torsion-free passage exists; determine their finiteness properties.

(5) Zeta integrality. For a number field $k$, is each value $w_{2i}\zeta_k(1-2i)$ an integer (individually, beyond the integrality of the alternating product that the Euler-Poincare formula provides)? Do Kummer-type congruences hold among these values, and do the ray-class zeta function analogues admit Euler-Poincare interpretations?

Of these, (1) is settled negatively, and the arithmetic cases of (2), the finiteness part of (4), and the odd-primary totally real part of (5) are settled. The surviving open core, stated at the source's generality, consists of the following four independently posed open questions: the general non-arithmetic lattice case of (2); the precise noncompact comparison statement of (3); an Euler-Poincare and integrality theory for not-virtually-torsion-free function-field S-arithmetic groups in (4); and the two-primary and general-field parts of (5) together with its Kummer-type congruences. The four parts are linked by their common objective, but none presupposes the others.

References

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